A modular approach to the generalized Ramanujan-Nagell equation

Abstract

Let k be a positive integer. In this paper, using the modular approach, we prove that if k ≡ 0 (mod 4), 30 < k < 724 and 2k − 1 is an odd prime power, then the equation ( * ) x 2 + (2k − 1) y = k z has only one positive integer solution (x, y, z) = (k − 1, 1, 2). The above results solve some difficult cases of Terai's conecture concerning the equation ( * ). 2020 Mathematics Subject Classification. 11D61.